A-level · Algebra & functions

Surds and indicesExam-style questions

Powers and roots, kept exact. The three index laws and why they hold; why a zero power is 1 and a negative power is "one over"; fractional powers as a root and a power; writing an expression as powers of x, ready for calculus; simplifying surds, and the root of a sum that does not split; and clearing a surd from a denominator with a conjugate.

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16 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.
(a) By substituting $p = 6^{x}$ , show that the equation

$36^{x+\frac{1}{2}} + 3 \times 6^{x+1} = 6^{x} + 3$

can be written in the form

$6p^{2} + 17p - 3 = 0$

(3 marks)
(b) Hence solve

$36^{x+\frac{1}{2}} + 3 \times 6^{x+1} = 6^{x} + 3$

(3 marks)
Level 4 · 6 marksTry it
2
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.
(a) Write

$\frac{11 + 5\sqrt{3}}{2 + \sqrt{3}}$

in the form $a + b\sqrt{3}$ where $a$ and $b$ are integers to be found.

(3 marks)
(b) Hence, or otherwise, solve

$2x + 3 = 14 + \sqrt{3}\,(5 - x)$

giving your answer in simplest form.

(3 marks)
Level 4 · 6 marksTry it
3
In this question you must show detailed reasoning.

Solutions relying on calculator technology are not acceptable.
(a) Find the exact solutions of the equation

$x^{3} + 7x = 6x^{2}$

(4 marks)
(b) Hence solve the equation

$y\sqrt{y} + 7\sqrt{y} = 6y \qquad y > 0$

giving each answer in the form $a + b\sqrt{2}$ , where $a$ and $b$ are integers.

(2 marks)
Level 4 · 6 marksTry it
4
In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.
(a) Given that $w = 2^{x}$ , write each of the following in the form $kw^{n}$ , where $k$ and $n$ are rational constants.

(i) $2^{3x - 2}$

(ii) $\dfrac{4^{x + 1}}{\sqrt{2^{x}}}$

(4 marks)
(b) (i) Show that, when $p = 8^{t}$ , the equation

$2^{6t + 2} + 2 = 2^{3t + 3} + 8^{t}$

becomes

$4p^{2} - 9p + 2 = 0$

(ii) Hence find the exact values of $t$ that satisfy this equation.

(5 marks)
Level 4 · 9 marksTry it
5
Simplify fully.
(a) $3p^2 \times 8p^{-\frac13}$

(2 marks)
(b) $\left(2q^{-3}\right)^4 \times (4q)^{\frac12}$

(2 marks)
(c) $7 \times 9^c \times 27^{2c} - 4 \times 3^{8c}$ , giving your answer as a single power of $3$ .

(4 marks)
Level 4 · 8 marksTry it